High resolution imaging and microscopy revolutionized various fields of science and technology. Over the past century extensive efforts have been directed towards enhancing imaging resolution as well as enabling imaging of objects of smaller dimensions. The main limitation in resolving mini-scale structures is the diffraction limit, dictated by the optical wavelength.
A significant advance in resolving nanoscale structures was offered with the recent development of new laser sources such as the free electron laser (FEL) [2, 3], which provide coherent radiation in the x-ray regime enabling a use of shorter wavelengths. An additional important advance was the realization that crystallographic methods can be applied to non periodic objects. This approach, leading to techniques known as Coherent Diffractive Imaging (CDI) [4] or lensless imaging, has already been successfully used for imaging a variety of objects, from a yeast cell [5] through nano-crystals [3] and even a single virus [2].
Lensless imaging techniques enable indirect observation of objects (e.g. molecular structure) with high resolution by measuring intensity of diffraction patterns generated by scattering of coherent light from the objects. As known, light diffracted from an object forms a diffraction pattern indicative of a Fourier transform of the object. Reconstruction of an image of the object requires retrieving the phase of the diffraction pattern and calculating an inverse Fourier transform. However, retrieval of the phase data from intensity measurements is not a trivial task. This problem is generally denoted as the non-crystallographic phase problem. Various techniques of phase retrieval from intensity measurements have been subject of intensive studies over the past decades and are still some of the major challenges in lensless imaging techniques.
Generally, most objects to be imaged have a finite extent, usually termed the “compact support” of the object. Mathematically, this is the region where the function has non-zero value. It has been shown that for such finite objects, the two dimensional phase retrieval problem has a unique solution up to trivial ambiguities (linear phase and conjugation, generally corresponding to lateral shifts and/or reflections). In practice however, retrieving the phase based on intensity measurements is still a major challenge.
Over the past decades numerous approaches were proposed to solve the phase retrieval problem. The currently available techniques are divided based on a natural tradeoff between experimental complexity and computational reconstruction efficiency. On one end, simple and direct measurements of the diffracted intensity require intensive and iterative computational phase retrieval [6, 7]. On the other, multiple measurements, with increased experimental complexity enable to reduce the computational complexity of the phase retrieval problem.
In general, and even under noise free measurements, the available iterative phase retrieval approaches suffer from several limitations. The iterative algorithms do not always converge, and even when the calculation converges to a certain solution, corresponding to a local minimum of some optimization functional, there is no assurance that this solution is the correct one. Additionally, there is no clear and robust technique to evaluate the effects of noise or to provide exact estimation of the compact support thereby providing a measure for an error in the calculated solution. These difficulties have been addressed by recent approaches introducing new constraints and thus requiring additional measurements. One example is the Ptychographical Iterative Engine (PIE) [8], in which the object is scanned with a small known aperture, keeping an overlap between the inspection regions of every two positions of the aperture.
Additional recent approaches [9, 10] utilize structured illuminations of the object to be observed, in order to reduce the phase retrieval problem to a convex one. Although these techniques are mathematically promising, they raise various experimental difficulties limiting their efficiency. Moreover, these methods are highly computationally demanding and thus are limited to small images.
Recently, a novel technique was introduced by the inventors of the present invention for 1D Vectorial Phase Retrieval (VPR) [11, 12]. Generally, the VPR technique is useful for phase retrieval in problems where the signal of interest has a vectorial, or vectorial-like, nature. Examples of such vectorial like signal include signals having polarization or spin states. According to this technique, if the two components of the signal (for example x and y polarization components) are independent, and the signal has a certain (possibly unknown) support, then the 1D VPR technique provides the unique solution. Additionally, the VPR technique offers a significant advantage by allowing to solve the phase problem with a set of linear equations, making it both scalable and robust to noise.